TESTING THE ASSUMPTIONS OF
VARIABLES CONTROL CHARTS AND AN
APPLICATION ON FOOD INDUSTRY
Berna YAZICI
Department of Statistics, Anadolu University
Eskisehir,Turkey
E-mail:[email protected]
ABSTRACT
In this study the statistical assumptions to construct
variables quality control charts have been held. Testing those
assumptions are mentioned and the solutions for the researcher, in
case of violation of the assumptions are also explained.
INTRODUCTION
Quality is generally desirable characteristics of a product or
service should have. The customers have many options to select a
product or service. So companies need improve the quality of they
produce to survive. Quality improvement is the reduction of
variability in processes and products. Variability is described by
statistical methods. Control charts are one of the tools that is used
to detect whether the process is under control or not. But those
charts may cause misunderstandings if the researchers make
decisions in case of violation some assumptions. These
assumptions
are
uncorrelated
measurements,
normality,
homoscedasticity and homogeneity of means.
INTRODUCTION
For application, 144 measurements are taken from a factory
that produces wafers. The thickness of wafers is in question for
statistical process control studies in this company. For the study 16
samples, with 9 measurements each, are used. All samples are
taken by the same worker half an hour periodically. Each of the
assumptions mentioned above has been tested on this data set.
Recommendations in case of violation of each assumption are
mentioned.
1. UNCORRELATED MEASUREMENTS ASSUMPTION
All the samples selected randomly are independent of the one immediately
preceding and the one immediately following, briefly independency of the
measurements.
• In Eq. L is the amount of lag. Lrn =
( X 1 X L +1 + X 2 X L + 2 + ... + X n X L ) - X
X2-X
X

• In runs test, duration of completed runs (d) is
important. The expected number of completed runs
of deviations is and the expected number of
completed runs of all durations is

X
n - 3
E (  fˆ ) =
2
fˆ =
n - d - 1
+
2 d 1
FOR THE DATA SET OF WAFER THICKNESS
• Lower point: –0.462 upper point is 0.328. The result is not between the
confidence interval limits
1 r16 = 0.54
• Runs test
c2
calc
= 45.71 >
c2
table
=2.167
We reject the null hypothesis of uncorrelated measurements
1. UNCORRELATED MEASUREMENTS ASSUMPTION
If the assumption is violated
• In this case researcher may fit an ARIMA model and apply standard control
charts to residuals instead of the raw data. Residuals will give uncorrelated
results.
• Exponentially weighted moving average (EWMA) control charts can be
used by moving centerline, with control limits based on prediction error
variance.
• To decide whether or not an autocorrelated process may be considered in
control, one must investigate the reasons for the autocorrelation. After that,
it will be easier to eliminate the autocorrelation by using an engineering
controller.
In this case, the reason of the autocorrelation can be
determined and uncorrelated measurements can be held.
• One way to remove autocorrelation is taking the samples in larger sampling
intervals if the process structure is suitable. In this way, Shewhart control
charts become appropriate.
2. NORMALITY ASSUMPTION
The distribution of means will be normal if the population is normally
distributed.
• c2 test of goodness of fit
• Shapiro-Wilk’s W test for normality
• where bi is calculated as follows mi representing
the expected values of the order statistics from a
unit normal distribution
 n
 2
  (b i X i )
i =1


=
W
 (X i - X ) 2
bi =
mi
n
 m i2
i =1
• Graphical methods that the researcher can apply using computer packages
for testing the normality, such as Q-Q Plot, Lilliefors Test
FOR THE DATA SET OF WAFER THICKNESS
c2
=
0.959
<
calc
table =1.635
c2
• Shapiro-Wilk’s W test for normality
Wcal = 0.9694 critical value for =0.05 and 16 from table is 0.887.
Wcal > W(16; 0.05)
We cannot reject the hypothesis of normality.
2. NORMALITY ASSUMPTION
If the assumption is violated
• The Camp-Meidell adjustment for normality can be made
• According to Tchebycheff inequality, no matter the
shape of the distribution
100 -
100
2.25z 2
100 -
100
z2
• If the population is not too skewed and unimodal larger sample sizes
suffice the normality assumption due to central limit theorem
• The development of X-bar, R and S chart mechanics is based on the
process metric being normally distributed. However, the
chart itself is
robust to deviations from normality depending on the central limit theorem
3. HOMOGENEITY OF VARIANCES
The variances within each of the samples must be equal
• One way to test homogeneity of variances is Cochran’s g test
• 0 test
0 =
2
sT
(s12 s 22 ...s 2m )1 / m
M
• Bartlett’s test c 2 =
c
 m 1 
1
1 


=
+


c 1

3(m - 1)  i =1 n i - 1  N - m 
sum of s 2
(s12 + s 22 + ... + s 2m ) / m
1 =
(s12 s 22 ...s 2m )1 / m
• 1 test
M = ( N - m) ln s 2p -
m
 (ni - 1) ln si2
i =1
m
 (n i - 1)s i2
=
s 2p = i 1
g=
l arg ests 2
N-m
FOR THE DATA SET OF WAFER THICKNESS
• 0 = 1.727 > table = 1.41
• 1=1.31 >table=1.23
• Bartlett’s test
c2
=
59.846
>
table
calc
c2
We reject the null hypothesis of equal variances.
3. HOMOGENEITY OF VARIANCES
If the assumption is violated
• Taking new samples with equal number of observations may be the best
solution
4. HOMOGENEITY OF MEANS
The control charts are constructed with the homogeny samples from a
process
• 0 test
0 =
2
sT
(s12 s 22 ...s 2m )1 / m
• ANOVA test
Before constructing an ANOVA test, one must be sure whether there
are extraordinary sample mean or not
X1+ i - X1
=
rij
X n - j - X1
F=
s 2B
s 2w
FOR THE DATA SET OF WAFER THICKNESS
• Critical value from Dixon’s table is 0.507>0.15 We cannot reject the H0
hypothesis and we conclude that there are not any extraordinary sample
mean among 16 sample means.
• Fcal = 2.566>F0.05;15;128=2.11
We conclude that the means are not homogenous
4. HOMOGENEITY OF MEANS
If the assumption is violated
• One can select new samples by equal time intervals
The methods described here are summarized by a flow chart on next four slides
Test uncorrelated
measurements
assumption
Use a test depending
on the circular
autocorrelation coefficient
Use runs test
Assumption
satisfied
Yes
No
Fit ARIMA model
Use EWMA
control charts
Research the reason
of autocorrelation
Test the normality
assumption
Take samples
in larger sample
intervals
Test the normality
assumption
Use c2 test
Use
Shopiro-Wilk’s W
test
Assumption
satisfied
Use
Graphical methots
Yes
No
Use
Camp-Meidell
adjustment
Use
Tchebycheff
inequality
Take larger samples
Test homogeneity
of variances
assumption
Only use
X-bar charts
Test homogeneity
of variances
assumption
Use
Cochran’s g test
Use
0
test
Use
1
Assumption
satisfied
No
Take new samples
with equal numbers
of observations
Test homogeneity
of means
assumption
Yes
test
Use
Bartlett’s test
Test homogeneity
of means
assumption
Use
ANOVA
test
Assumption
satisfied
Yes
No
Take new
samples by
equal time intervals
Construct the
control charts
CONCLUSIONS AND RECOMMENDATIONS
In statistical process control studies, variables control charts are one of the
best guide for the researcher to detect the changes in the process. Before
constructing those charts firstly some assumptions must be tested. The
assumptions in question are uncorrelated measurements, normality,
homogeneity of variances and homogeneity of means. To avoid the
misunderstandings and wrong interpretations of these charts, one should
test those assumptions and if the assumptions are satisfied, then the charts
must be constructed.
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[2]
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[3]
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[4]
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[5]
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[6]
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[7]
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[8]
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[9]
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